| The theory of stochastic differential equations(SDEs) has been in existence for nearly sixty years now. Complicated finite dimensional stochastic dynamics can now be modelled and understood theoretically through the Ito stochastic calculus and the more general theory of semimartingales. SDEs now find application as a matter of course in diverse disciplines such as in civil and mechanical engineering, economics and finance, environmental science, signal processing and filtering, chemistry and physics, population dynamics and psychology, pharmacology and medicine, to mention just a few. Stochatic differential equations (SDEs) arise in mathematical models of physical systems that possess inherent noise and uncertainty. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics and finance. Using stochastic differential equations we can successfully model systems that function in the presence of random perturbations. Such systems are among the basic objects of modern control theory.Many applications require the computation of a functional of a solution of an SDE such as a moment or the expectation of a terminal pay-off. Numerically this is easier because only a probability measure of the solution has to be approximated rather that its highly irregular sample paths, but there are nevertheless still many practical complications. Higher order schemes for this weak kind of convergence can also be constructed systematically from stochastic Taylor expansions, but now the driving noise integrals can be approximated by simpler, more easily generated random variables depending on the desired order of the scheme.Other numerical methods are also available for the computations of such functionals, which are often the solution of a partial differential equation(PDE) related to the Kol-mogorov backward or Fokker- Planck equation. In principle, standard finite-difference...
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